At the top right of the Hangout conversation window, click the gear icon.

Select Delete Hangout.

Click Delete.

Delete history for a group Hangout

Currently, there’s no way to delete the Hangout history for a group Hangout and still stay in the Hangout. If you want to clear the Hangout history, you’ll need to leave the Hangout completely. When you leave the Hangout, your message history will be deleted for only you – others in the Hangout will still see your message history. Also, you’ll no longer receive new message notifications.

Your Hangout history may be saved as several email threads. Deleting your message history will delete all email threads associated with that Hangout. Deleted messages won’t be stored in your Trash folder.

Moving messages to the trash in Gmail:

Your Hangout may be stored as a series of email threads within Gmail. Moving the threads to the trash will only remove the messages from your Gmail mailbox. Moving the email threads won’t delete the messages from the Hangout window or on the Hangouts mobile app.